Comptes Rendus
Algebraic geometry
The connectedness of degeneracy loci in positive characteristic
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 959-964.

A well-known result of Fulton–Lazarsfeld ensures the connectedness of degeneracy loci under an ampleness condition. We extend it to positive characteristic, along with the variants for degeneracy loci of symmetric and alternating maps of even rank, due to Tu in characteristic zero. The proof uses the explicit determination of the top étale cohomology group of an algebraic variety, a result communicated by Esnault.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/crmath.448
Classification: 14F20, 14N05, 14J60, 14F06, 14M12, 14F45, 14E15, 14G17

Rémi Lodh 1

1 Springer-Verlag, Tiergartenstr. 17, 69121 Heidelberg, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2023__361_G6_959_0,
     author = {R\'emi Lodh},
     title = {The connectedness of degeneracy loci in positive characteristic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {959--964},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.448},
     language = {en},
}
TY  - JOUR
AU  - Rémi Lodh
TI  - The connectedness of degeneracy loci in positive characteristic
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 959
EP  - 964
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.448
LA  - en
ID  - CRMATH_2023__361_G6_959_0
ER  - 
%0 Journal Article
%A Rémi Lodh
%T The connectedness of degeneracy loci in positive characteristic
%J Comptes Rendus. Mathématique
%D 2023
%P 959-964
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.448
%G en
%F CRMATH_2023__361_G6_959_0
Rémi Lodh. The connectedness of degeneracy loci in positive characteristic. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 959-964. doi : 10.5802/crmath.448. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.448/

[1] Emil Artin Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, 3, Interscience Publishers, 1957

[2] Pierre Deligne Exposé XVIII: La formule de dualité globale, Théorie des topos et cohomologie étale des schémas (SGA 4), tome 3 (Michel Artin; Alexander Grothendieck; Jean-Louis Verdier; Pierre Deligne; Bernard Saint-Donat, eds.) (Lecture Notes in Mathematics), Volume 305, Springer, 1973 | DOI | Zbl

[3] Hubert Flenner; Bernd Ulrich Codimension and connectedness of degeneracy loci over local rings, Math. Z., Volume 286 (2017) no. 1-2, pp. 723-740 | DOI | MR | Zbl

[4] William Fulton; Robert Lazarsfeld On the connectedness of degeneracy loci and special divisors, Acta Math., Volume 146 (1981), pp. 271-283 | DOI | MR | Zbl

[5] William Fulton; Robert Lazarsfeld Positive polynomials for ample vector bundles, Ann. Math., Volume 118 (1983), pp. 35-60 | DOI | MR | Zbl

[6] Alexander Grothendieck Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publ. Math., Inst. Hautes Étud. Sci., Volume 32 (1967), pp. 5-361 (written in collaboration with J. Dieudonné) | Numdam | Zbl

[7] Alexander Grothendieck; Jean A. Dieudonné Éléments de géométrie algébrique I, Grundlehren der Mathematischen Wissenschaften, 166, Springer, 1971

[8] Robin Hartshorne Ample vector bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 29 (1966), pp. 63-94 | Numdam | Zbl

[9] Luc Illusie; Michael Temkin Exposé X. Gabber’s modification theorem (log smooth case), Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École Polytechnique 2006–2008 (Luc Illusie; Yves Laszlo; Fabrice Orgogozo, eds.) (Astérisque), Société Mathématique de France, 2014 no. 363-364, pp. 167-212 | Zbl

[10] Aise J. de Jong Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 51-93 | Numdam | Zbl

[11] Robert Lazarsfeld Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer, 2004

[12] Winfred Scharlau Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, 270, Springer, 1985 | DOI

[13] Siu-Kei Tin Numerically positive polynomials for k-ample vector bundles, Math. Ann., Volume 294 (1992) no. 4, pp. 579-590 | MR | Zbl

[14] Loring W. Tu The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks, Trans. Am. Math. Soc., Volume 313 (1989) no. 1, pp. 381-392 | MR | Zbl

[15] Loring W. Tu The connectedness of degeneracy loci, Topics in algebra. Part 2: Commutative rings and algebraic groups (Banach Center Publications), Volume 26, PWN-Polish Scientific Publishers, 1990 no. 2, pp. 235-248 | MR | Zbl

Cited by Sources:

Comments - Policy